Lesson 9: Side Length Quotients in Similar Triangles

Side Length Quotients in Similar Triangles

Student Summary

If 2 polygons are similar, then the side lengths in one polygon are multiplied by the same scale factor to give the corresponding side lengths in the other polygon.

For these triangles the scale factor is 2:

<p>Two triangles. First, A, B C. Length A, B, 5, length B C, 4, length C A, 3. Second, A, prime B prime C prime. Length A, prime B prime, 10, B prime C prime 8, C prime A, prime.</p><br>

Here is a table that shows relationships between the lengths of the short and medium sides of the 2 triangles.

	small triangle	large triangle
medium side	4	8
short side	3	6
(medium side) $\div$ (short side)	$\frac{4}{3}$	$\frac{8}{6} = \frac{4}{3}$

The lengths of the medium side and the short side are in a ratio of $4:3$ . This means that the medium side in each triangle is $\frac43$ as long as the short side. This is true for all similar polygons: the ratio between 2 sides in one polygon is the same as the ratio of the corresponding sides in a similar polygon.

We can use these facts to calculate missing lengths in similar polygons. For example, triangles $ABC$ and $A’B’C’$ are similar.

Since side $BC$ is twice as long as side $AB$ , side $B’C’$ must be twice as long as side $A’B’$ . Since $A’B’$ is 1.2 units long and $2\boldcdot1.2=2.4$ , the length of side $B’C’$ is 2.4 units.

<p>Two triangles. First, A, B C. Length A, B, 3, length B C,6. Second, A, prime B prime C prime. Length A, prime B prime, 1 point 2.</p><br>

Visual / Anchor Chart

Standards

Building On

7.RP.2

Recognize and represent proportional relationships between quantities.

Addressing

8.G.A

No additional information available.

8.G.4

Know that a two-dimensional figure is similar to another if the corresponding angles are congruent and the corresponding sides are in proportion. Equivalently, two two-dimensional figures are similar if one is the image of the other after a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence that maps the similarity between them on the coordinate plane.